For a function $f$ and a probability measure $P$ and $\forall \delta > 0, \lambda \in \mathbb{R}$ it is apparently true that, $$P(\vert f - \mathbb{E}_P [f]\vert > \delta ) \leq e^{-\lambda \delta }\mathbb{E}_P [e^{\lambda \vert f - \mathbb{E}_P [f] \vert } ]$$
- This is apparently Chebyshev's inequality. Can someone kindly explain how?
If necessary assume that $P$ satisfies a Log Sobolev Inequality and $f$ is Lipschitz.